So, P is false and Q is false in your case. Thus, the statement is true.

Apr 22nd 2009, 10:10 PM

Gamma

Vacuously True

This statement is an example of something that is vacuously true. Since the hypothesis is always false and will never be met, you could put literally anything you want after the then and it would be true.

If 1+1=3, then I am a purple cow. Sure this is true because .

Another way to think about this would be to take the contrapositive of the statement and this is logically equivalent to the original statement and you see this is:
If , then .

The first is true and the second statement is true, so the statement is true. Notice because the second statement is always true it doesn't matter what the first statement is, so these two methods of analysis agree.